Internal problem ID [11534]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 3, Laplace transform. Section 3.4 Impulsive sources. Exercises page
173
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_linear, `class A`]]
\[ \boxed {x^{\prime }+3 x=\delta \left (t -1\right )+\operatorname {Heaviside}\left (t -4\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 38
dsolve([diff(x(t),t)+3*x(t)=Dirac(t-1)+Heaviside(t-4),x(0) = 1],x(t), singsol=all)
\[ x \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}+\frac {\operatorname {Heaviside}\left (t -4\right )}{3}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{3}+{\mathrm e}^{-3 t} \]
✓ Solution by Mathematica
Time used: 0.201 (sec). Leaf size: 53
DSolve[{x'[t]+3*x[t]==DiracDelta[t-1]+UnitStep[t-4],{x[0]==1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {1}{3} e^{-3 t} \left (3 e^3 \theta (t-1)+\left (e^{12}-e^{3 t}\right ) \theta (4-t)+e^{3 t}-e^{12}+3\right ) \]